Transcript of "Ds33717725"

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Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725717 | P a g eOptimal Gear Design By Using Box And Random SearchMethodsAnitha Santhoshi.M*, Durga Devi.G***(Asst.prof, Department of Mechanical Engg, SVCET, Etcherla)** (Asst.prof, Department of Mechanical Engg, SVCET, Etcherla)ABSTRACTThe development of evolutionary algorithmsplays a major role, in recent days, for optimaldesign of gears, so as to reduce the weight. In thisstudy an optimal weight design (OWD) problemof gear is formulated for constrained bendingstrength of gear, tortional strength of shafts andeach gear dimension as a NIP problem and solvedit directly by keeping nonlinear constraint usingBox and Random search methods, such that thenumber of decision {design} variables does notincrease and easily get the best compromisedsolution. An extensive computer program in Javahas been written exclusively for their purposeand is successfully used to obtain the optimal geardesign.Keywords – optimal weight design (OWD), NIPproblem, Box Method, Random search method,Decision {design} variables.1. INTRODUCTIONThe most important problem that confrontspractical engineers is the mechanical design, a fieldof creativity. In case of gear design, an infinitenumber of possible design solutions are found withinthe overall objective. Any one of these solutions isadequate because it represents a synthesis, whichmerely satisfies the functional requirements [1].Here lies a conductive environment for applying cutand try technique to obtain an optimal designsolution among the available solutions. Theapproach to solve certain design problem has sorelied on the trail-and-cut methods which, because oftheir methodology, take considerable time to obtainthe optimal solution.In this study an optimal weight design(OWD) problem of gear is formulated forconstrained bending strength of gear, tortionalstrength of shafts and each gear dimension as a NIPproblem and solves it directly by keeping nonlinearconstraint by using Box and Random SearchMethods.As a result, the number of decision (design)variables does not increase and easily get the bestcompromised solution. An extensive computerprogram in Java has been written exclusively fortheir purpose and is successfully used to obtain theoptimal gear design.2. ENGINEERING OPTIMIZATIONOptimization is the act of obtaining the bestresult under given circumstances. In design,construction and maintenance of any engineeringsystem, engineers have to take many technologicaland managerial decisions at several stages. Theultimate goal of all such decisions is either tominimize the effort required or to maximize thedesired benefit.2.1 Optimization Algorithms2.1.1 Single variable optimization algorithmsThese algorithms provide a goodunderstanding of the properties of the minimum andmaximum points in a function and how optimizationalgorithms work iteratively to find the optimumpoint in a problem. The algorithms are classified intotwo categories, they are direct methods and gradientbased methods. Direct methods do not use anyderivative information of the objective function:only objective function values are used to guide thesearch process. However, gradient based methodsuse derivative information (first and/ or secondorder) to guide search process.2.1.2 Multi – variable optimization algorithmsA number of algorithms for unconstrained,multi-variable optimization problems are present.These algorithms demonstrate how this search foroptimum points progress in multiple dimensions.2.1.3 Constrained optimization algorithmsConstrained optimization algorithms usedin single variable and multi variable optimizationalgorithms repeatedly and simultaneouslymaintained the search effort inside the feasiblesearch region. These algorithms are mostly used inengineering optimization problems. Thesealgorithms are divided into two broad categories;they are direct search methods and gradient-basedmethods. In constraint optimization problem,equality constraints make the search process slowand difficult to converge.

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Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725718 | P a g e2.1.4 Specialized optimization algorithmsThere exist a number of structuredalgorithms, which are ideal for only a certain class ofoptimization problems. Two of these algorithms areinteger programming and geometric programming.These are often used in engineering designproblems. Integer programming methods can solveoptimization problems with integer design variables.Geometric programming methods solve optimizationproblems with objective functions and constraintswritten in a special form.3. BOX METHODThe Box method is similar to the simplexmethod of unconstrained search except that theconstraints are handled in the former method. Thismethod was developed by M.J.Box in 1965; [2], thealgorithm begins with a number of feasible pointscreated at random. If a point is found to beinfeasible, a new point is created using thepreviously – generated feasible points. Usually, theinfeasible point is pushed towards the centroid of thepreviously found feasible points. Once a set offeasible points is found, the worst point is reflectedabout the centroid of rest of the points to find a newpoint, Depending on the feasibility and functionvalue of the new point, the point is further modifiedor accepted. If the new point falls outside thevariable boundaries, the point is modified to fall onthe violated boundary. If the new point is infeasible,the point is retracted to towards the feasible points.The worst point in the simplex is replaced by thisnew feasible point and the algorithm continues forthe next iteration. The Box Method is also called as“Complex Search Method”.3.1 BOX [Complex Search] Algorithm [2]Step 1: Assume a bound in x (x (L), x (U)), areflection parameter α.Step 2: Generate an initial set of P (usually 2n)feasible points. For each point(a) Sample n times to determine the point)( pix in the given bound.(b) If x (p)is infeasible, calculate x(centroid) of current set of points and reset)(21 )()()( pppxxxx Until)( px is feasible;Else if)( px is feasible, continue with (a)until P points are created(c) Evaluate )( )( pxf for p = 0, 1, 2…, (P-1)Step 3: Carry out the reflection step:(a) Select xRsuch thatf (xR) = max f(x (p)) = Fmax(b) Calculate the centroi x d (of pointsexcept xR) and the next point)( Rmxxxx  (c) If xmis feasible and f (xm) > Fmaxretract half the distance to thecentroid x . Continue until f(xm) < FmaxElse if xmis feasible and f (xm) < Fmax,go to Step 5.Else if xmis infeasible, go to Step 4.Step 4: Check for feasibility of the solution(a) For all i, reset violated variablebounds:If)()( LimiLimi xxsetxx If)()( UimiUimi xxsetxx (b) If the resulting xmis infeasible, retracthalf the distance to the centroid.Continue until xmis feasible. Go toStep 3(c).Step 5: Replace xRby xm. Check for termination.Calculate ppxfPf )(1 )(and x =ppxP)(1

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Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725720 | P a g e4. RANDOM SEARCH METHODLike the Box method, the random searchmethod also works with a population of points. Butinstead of replacing the next point in the populationby a point created in a structured manner points arecreated either at random or by performing aunidirectional search along the random searchdirections. Here, we describe one such method.Since there is no specific search direction used in themethod, random search methods work equallyefficiently to many problems. In the Luus andjaakola method 1973; an initial point and an initialinterval are chosen at random. Depending on thefunction values at a number of random points in theinterval, the search interval is reduced at everyiteration by a constant factor. Then it is increased. Inthe following algorithm, P points are considered ateach iteration and Q such iterations are performed.Thus, if the initial interval in one variable is do andat every iteration the interval is reduced by a factor€, the final accuracy in the solution in that variablebecomes (1-€) ^Q do and the required number offunction evaluations is P X Q.4.1. Random Search Algorithm [2]Step 1 Given an initial feasible point x0,an initial range zosuch that the minimum, x*, lies in)21,21( 0000zxzx  Choose the Parameter0< 1 For each of Q blocks, initially set q = 1 &p = 1.Step 2 For i = 1, 2 …N, create pointsusing a uniform distribution of r in the range (-0.5,0.5). Set11)(  qiqipi rzxxStep 3 If)( px is infeasible and p < P, repeatStep-2. If x(p)is feasible, save x(p)and f(x(p)),increment p and repeat Step-2;Else if p = P, set xqto be the point that has thelowest f(x(p)), overall feasible x(p)including xq-1andreset p = 1.Step 4 Reduce the rangevia1)1(  qiqi zz .Step 5 If q > Q, Terminate;Else increment q and continue with Step-2.The suggested values of parameters are €=0.05, P= 5(depending upon the design variables),and Q is related to the desired accuracy in thesolution. It is to be noted that the obtained solution isnot guaranteed to be the true optimum.5. PROBLEM DESCRIPTIONIn the present work an OWD of a gear witha minimum weight is considered in fig above. Inputpower of 7.5 KW, the speed of crank shaft gear(pinion) is considered to be 1500 rpm and the gearratio is 4. Necessary conditions required fordeveloping a mathematical model for gear design arediscussed in this section as given in [4].Preliminary Gear considerations: The followingare input parameters required for preliminary geardesign [6].1. Power to be transmitted (H), KW.2. Speed of the pinion (N1), rpm.3. Gear ratio (a)